Astronomical Distances or Measuring the Universe (Chapters 5 & 6) by Rastorguev Alexey, professor of the Moscow State University and Sternberg Astronomical Institute, Russia. Sternberg Astronomical Institute Moscow University. Content.
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Astronomical Distancesor Measuring the Universe(Chapters 5 & 6)by Rastorguev Alexey,professor of the Moscow State University and Sternberg Astronomical Institute, Russia
Sternberg Astronomical
Institute
Moscow University
Main-Sequence Fitting, or
the distance scale of star clusters
but of different initial masses
-----------------------------------------------
with accurately measured spectral energy distributions (SED),
or calibration based on the principles of the “synthetic photometry”
>1
BCV≤ 0
By definition,Mbol = MV + BCV
Example:BCV vs lg Teff:unique relation for all luminosities
From P.Flower (ApJ
V.469, P.355, 1996)
lgTeff - BCV – (B-V) from observations
Multicolor synthetic-photometry approach;
lgTeff–BCV–(U-B)-(B-V)-(V-I)-(V-K)-…-(K-L),
for dwarf and giants with [Fe/H]=+1…-3
(with step 0.5 in [Fe/H])
Pleiades problem:
HST gives smaller
parallax (by ~8%)
ΔMHp≈ -0.17m
MHp
(V-I)
Primary empirical calibration of the Hyades MS & isochrone for different colors, by HIPPARCOS parallaxes(M.Pinsonneault et al. ApJ V.600, P.946, 2004)
Solid line: theoretical isochrone with
Lejeune et al. (A&AS V.130, P.65, 1998)
color-temperature calibrations
MV
ZAMS
Giants
Turn-off point
Turn-off point
Turn-off point
0.1
1
10 Gyr
0.1
1
What are fancy shapes !
1 Gyr
ZAMS
ZAMS
ZAMS
H: Δ (color) = CE (color excess)
V: (m-M) = (m-M)0 + R· CE
(m-M)0 – true distance modulus
You can start writing paper !
Magnitudes offset
gives
ΔV=(V-MV)0+RV∙E(B-V)
↨
(m-M)0 = 5.60
E(B-V)=0.04
lg (age) = 8.00
ZAMS
G.Meynet et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
(m-M)0=12.55
E(B-V)=0.38
lg (age)=7.15
G.Meynet et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
(m-M)0=13.65
E(B-V)=0.56
lg (age)=7.15
RSG
(Red
Super-
Giants)
G.Meynet et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
RSG
(m-M)0=12.10
E(B-V)=0.32
lg (age)=8.22
G.Meynet et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
(m-M)0=7.88
E(B-V)=0.02
lg (age)=9.25
G.Meynet et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
(m-M)0=9.60
E(B-V)=0.03
lg (age)=9.60
G.Meynet et al.
(A&AS V.98, P.477,
1993)
Geneva isochrones
(kinematic selection; reason –
small velocity dispersion)
Field stars
Cluster stars
Δ(m-M)0≈±0.1m(random) ± 0.2m(systematic)
Typical distance accuracy of remote open clusters is ~10-15%
indicators, including
RR Lyrae variables
(with nearly constant
luminosity, see later)
RR Lyrae
BHB
(EHB)
TP
more metal-deficient
u 3551Å
g 4686Å
r 6165Å
i 7481Å
z 8931Å
Å
Isochrones fitting
example: M92
Age step 2 Gyr
Theoretical background of
this method is quite
straightforward
Galactic Globular Clusters
are distant objects and
very difficult to study,
even with HST
Reliable photometric data
exist mostly for brightest
stars: Horizontal Branch,
Red Giant Branch and
SubGiants
M.Salaris &
S.Cassisi,
“Evolution of stars
and stellar
populations”
(J.Wiley &
Sons, 2005)
Gyr
“Horizontal”
method
calibrations:
Color offset
vs [Fe/H]
for different
ages
Gyr
Gyr
Gyr
“Vertical” method calibrations: magnitude difference vs [Fe/H] for different ages
Gyr
ω Cen
Multiple populations ?
He abundance
differences ?
NGC 1806
(LMC)
Statistical parallaxes
VT=k r μ
VR
Sun
r
If all accepted distances are
systematically larger (shorter)
than true distances, then overestimated (underestimated)
tangential velocities will generally distort the ellipsoidal distribution of residual velocities, and the velocity ellipsoids will look like
… instead of being alike and pointed to the galactic center
only brief description follows
Photometric distances are calculated by star’s apparent and absolute magnitudes. Absolute magnitudes are affected by random and systematic errors. The last can be treated as systematic offset of distance scale used, ΔM.
Statistical parallax technique distinguishes:
expected distancere, calculated by accepted mean absolute magnitude of the sample (after luminosity calibration);
refined distancer, calculated by refined mean absolute magnitude of the sample (after application of statistical parallax technique);
true distancert, appropriate to true absolute magnitude of the star (generally unknown).
Toy distribution
of accepted and
refined absolute
magnitudes
ΔMV
Refined
mean
Excpectedandrefined absolute magnitudes (distances) differ due to systematic offset of the absolute magnitude, ΔMV, just what we have to found
Trueandrefined absolute magnitudes differ due to random factors (chemistry, stellar rotation, extinction, age etc.). Random scatter can be described in terms of absolute magnitude standard (rms) variance,σM
Expected
mean
True MV
Four components of 3D-velocity:
L = <ΔV·ΔV T>, T –transposition sign
Vloc (re) is what we measure !
Vb
VR
Vl
re
Galactic disk
Sun
Covariance tensor
Observed errors
Ellipsoidal distribution
Systematic motion: (a) relative to the Sun and (b) rotation
where
Individual velocities of all stars are independent on each other; in this case full (N-body) distribution function is the product of N individual functions f,
where N is the number of stars, A is the “vector” of unknown parameters to be found. Maximum Likelihood principle states that observed set of velocity differences is most probable of all possible sets. The set of parameters,A, is calculated under assumption that F reaches its maximum (or minimum, for maximum-likelihood functionLF )
Here|L| is matrix determinant, L-1 is inverse matrix. By minimizing LF by A, we calculate all important parameters {A}, for example:
Astronomical background:
A, Oort constant, derived from proper motions alone, depends on the distance scale used, whereas A, derived only from radial velocities, do not depend on the distances
As a result, scale factor can be estimated by requirement that both A values are equal to each other
Differential rotation contribution to space
velocity components in local approximation
r << R0(or|RP–R0 | << R0):
To first order by the ratio r/R0
in the expansion for the angular velocity:
From 1st Bottlinger equation (for radial velocity)
calculate contribution of the differential rotation toVr :
Linearity onr,
“double wave” onl
AOort’s constant (definition)
From 2nd Bottlinger equation (for velocity on l)
calculate contribution of the differential rotation toVl :
Linearity onr,
“double wave” onl
A0Vrdepends on the distance scale: A0Vr~ p -1
(decreases with increasing distances)
A0μldo not depend on the distance scale,A0μl≈const
The requirementA0Vr(p) ≈A0μl– robust method of the
adjusting the scale factor p
AVr
Optimal value of
the scale factor
Aμl